The effects of internal constraints on the configurations of chain molecules

We explore the three-dimensional configurations of chain molecules containing one or more self-contacts (constraints). We focus predominantly on the 1-, 2-, and 3-constraint ensembles. We take into account excluded volume by exhaustive computer enumeration of the conformational spaces of short chains on three-dimensional simple cubic lattices, and through use of the path integral approach of Edwards and Freed. We develop topological correlation functions to describe how the cyclization probability of one loop affects cyclization of another. There are two rather striking findings. (i) Considerable amounts of internal architecture (helices and antiparallel and parallel sheets) are predicted to arise in compact polymers due simply to steric restrictions. This appears to account for why there is so much internal organization in globular proteins. (ii) Several cyclization properties are remarkably ideal for chains which are relatively or highly compact in three dimensions. For example, in relatively compact molecules the correlation functions of loop pairs are well predicted by the random-flight model of Jacobson and Stockmayer; the number of configurations of maximally compact chains is predicted relatively well by the Flory theory of excluded volume, which is found to be better than the Huggins theory in three dimensions; and the probability of cyclization within globular chains is well predicted by the Bragg–Williams approximation.